In digital signal processing of audio and images, filter processing is frequently used. As the filter used for the filter processing, due to the feature that a linear phase is obtained with a finite number of taps, a linear phase FIR (Finite Impulse Response) filter is frequently utilized.
FIG. 1 is a diagram showing the configuration of a transversal type circuit of the linear phase FIR filter.
This linear phase FIR filter 1 has, as shown in FIG. 1, (n-1) number of delay units 2-1 to 2-n-1 cascade connected with respect to an input terminal TIN and configuring a shift register, n number of multipliers 3-1 to 3-n for multiplying filter coefficients h(0) to h(n-1) with respect to the signal input to the input terminal TIN and output signals of the delay units 2-1 to 2-n-1, and an adder 4 for adding the output signals of the n number multipliers 3-1 to 3-n and outputting the result to an output terminal TOUT.
As a representative design method of such a linear phase FIR filter, for example a Remez Exchange algorithm applied to a linear phase FIR filter by Parks, T. W. and McClellan, J. H. et al. is known (refer to for example Non-Patent Document 1).
The Remez Exchange algorithm is an algorithm for approximation so that a weighted approximation error exhibits an equal ripple shape with respect to a desired amplitude characteristic.
As applications of filter processing using a linear phase FIR filter, there are conversion of resolution of an image utilizing sampling rate conversion and conversion of the sampling frequency of audio.
For example, in conversion of resolution, use is made of a multi-rate filter using an interpolator, a decimeter, and a linear phase FIR filter as element technologies (refer to for example Non-Patent Document 2).
In a multi-rate filter, generally the linear phase FIR filter is used poly-phase decomposed matching with the interpolator. Both of the interpolator and the decimeter are cyclical time-invariant systems and have characteristic features different from that of a time-invariant system.
Due to the cyclical time invariability of the interpolator, distortion on a lattice called as “chessboard distortion” occurs in the conversion of resolution of the image.
Therefore, Harada and Takaie considered conditions for avoiding the chessboard distortion from a zero point arrangement of the filter (refer to Non-Patent Document 3).
A transmission function H(z) of the multi-rate filter not accompanied by chessboard distortion is found by multiplying a transmission function K(z) of the linear phase FIR filter (hereinafter referred to as an equalizer) designed by some sort of method by a zero point transmission function Z(z) in order to avoid the chessboard distortion later.
(Equation 1)H(z)=Z(z)·K(z)  (1)
(Equation 2)Z(z)=1+z−1+Z−2+ . . . +Z−(U-1)  (2)
Here, a previously fixed linear phase FIR filter like the zero point transmission function Z(z) for avoiding the chessboard distortion will be referred to as a pre-filter.
FIGS. 2A to 2C show an example of the frequency response of a multi-rate filter avoiding chessboard distortion by multiplying the equalizer designed by the Remez Exchange algorithm by a pre-filter and the weighted approximation error.
Non-Patent Document 1: Parks, T. W. and McClellan, J. H.: “Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase”, IEEE Trans. Circuit Theory, CT-19, 2, pp. 189-194, 1972, and Rabiner, L. R., McClellan, J. H. and Parks, T. W.: “FIR Digital Filter Design Techniques Using Weighted Chebyshev Approximation”, Proc. IEEE, Vol. 63, April, pp. 595-610, 1975;
Non-Patent Document 2: Takaie, Hitoshi, Multi-rate Signal Processing, Shokodo, 1997;
Non-Patent Document 3: Harada, Yasuhiro and Takaie, Hitoshi: Multi-rate Filter not Accompanied by Chessboard Distortion and Zero Point Arrangement of Same, Shingaku Giho CAS96-78, pp. 1-6, 1997-01